occurs so frequently in physics that it is not necessary to enumerate examples.
Here, is usually some sort of displacement or perturbation, whereas is the
(constant) wave speed. The wave equation possesses the formal solution

(231)

where and are arbitrary functions. The above solution represents arbitrarily
shaped wave pulses propagating with speed in the and directions, respectively, without
changing shape.

The wave equation, which is second-order in space and time,
can be written as two coupled first-order equations
by defining the new variables
and
.
Expressing Eq. (230) in terms of these new variables, we obtain

(232)

(233)

Note that when solving the wave equation numerically it is generally preferable to write it
as a set of coupled first-order equations, as shown above.